1 Introduction

Luo et al. [1] presented a comprehensive study of MPEC. Flegel and Kanzow [2] obtained short and elementary proof of the optimality conditions for MPEC using the standard Fritz-John conditions. Further, Flegel and Kanzow [3] introduced a new Abadie-type constraint qualification and a new Slater-type constraint qualification for the MPEC and proved that new Slater-type CQ implies new Abadie-type CQ. Ye [4] considered MPEC and introduced various stationary conditions and established that it is sufficient for being globally or locally optimal under some generalized convexity assumption and obtained new constraint qualifications.

Outrata et al. [5] derived necessary optimality conditions for those MPECs which can be treated by the implicit programming approach and proposed a solution method based on the bundle technique of nonsmooth optimization. Flegel et al. [6] considered optimization problems with a disjunctive structure of the feasible set and obtained optimality conditions for disjunctive programs with application to MPEC using Guignard-type constraint qualifications. Movahedian and Nobakhtian [7] introduced nonsmooth strong stationarity, M-stationarity and generalized the Abadie and Guignard-type constraint qualifications for nonsmooth MPEC. Movahedian and Nobakhtian [8] introduced a nonsmooth type of the M-stationary condition based on the Michel-Penot subdifferential and established the Fritz-John-type, Kuhn-Tucker-type M-stationary necessary conditions for the nonsmooth MPEC. Further, Movahedian and Nobakhtian [9] established necessary optimality conditions for Lipschitz MPEC on Asplund space and sufficient optimality conditions for nonsmooth MPEC in Banach space. We refer to the recent results of Ardali et al. [10], Chieu and Lee [11], Guo and Lin [12], Guo et al. [13, 14] and Ye and Zhang [15], and the references therein for more details related to the MPEC.

Following Luo et al. [1] and Movahedian and Nobakhtian [9], we consider the following mathematical programming problem with equilibrium constraints (MPEC):

$$\begin{aligned} (\mathrm{MPEC}) \quad \min& f(z) \\ \mbox{subject to: }& g(z)\leq0, \quad h(z)=0, \\ &G(z)\geq0,\qquad H(z)\geq0, \qquad \bigl\langle G(z),H(z) \bigr\rangle =0, \end{aligned}$$

where X is a Banach space, \(f:X \rightarrow\mathbb{R}\) is a lower semi-continuous (lsc) function, \(g:X \rightarrow\mathbb{R}^{k}\), \(h:X \rightarrow\mathbb{R}^{p}\), \(G:X \rightarrow\mathbb{R}^{l}\) and \(H: X \rightarrow\mathbb{R}^{l}\) are functions with lsc components.

The use of equilibrium constraints in modeling process engineering problems is a relatively new and exciting field of research; see Raghunathan and Biegler [16]. Hydroeconomic river basin models (HERBM) based on mathematical programming are conventionally formulated as explicit aggregate optimization problems with a single, aggregate objective function. Britz et al. [17] proposed a new solution format for hydroeconomic river basin models, based on a multiobjective optimization problem with equilibrium constraints, which allowed, inter alia, to express spatial externalities resulting from asymmetric access to water use.

Wolfe [18] formulated a dual program for a nonlinear programming problem. Motivated by a specific problem, namely the mathematical description of the rotating heavy chain, Toland [19, 20] introduced the notion of duality and established the duality theory for non-convex optimization problems. Rockafellar [21, 22] studied fundamental duality theory for convex programs using a conjugate function and established a generalized version of the Fenchelís duality theorem. In the last four decades there has been an extensive interest in the duality theory of nonlinear programming problems; see Mangasarian [23] and Mishra and Giorgi [24].

To the best of our knowledge, the dual problem to a nonsmooth MPEC has not been given in the literature as yet.

In this paper, we introduce Wolfe-type and Mond-Weir-type dual programs to the nonsmooth MPEC. We have established weak and strong duality theorems relating the nonsmooth MPEC and the two dual programs. The paper is organized as follows: in Section 2, we give some preliminaries, definitions, and results. In Section 3, we derive weak and strong duality theorems relating to the nonsmooth MPEC and the two dual models under convexity and generalized convexity assumptions.

2 Preliminaries

In this section, we give some notations, basic definitions, and preliminary results, which will be used later in the paper.

The Clarke-Rockafellar subdifferential of f is defined by

$$\partial_{c} f(x)= \bigl\{ x^{\ast}\in X^{\ast}: \bigl\langle x^{\ast}, v \bigr\rangle \leq f^{\uparrow}(x;v), \forall v \in X \bigr\} , $$

where

$$ f^{\uparrow}(x;v)=\sup_{\epsilon>0}\mathop{\mathop{\inf _{\gamma>0}}_{\delta>0}}_{\lambda>0} \mathop{\mathop{\sup _{y\in B(x;\gamma)}}_{f(y)\leq f(x)+\delta}}_{t\in(0,\lambda)} \inf_{w\in B(v;\epsilon)} \frac{f(y+tw)-f(y)}{t} $$

is the Clarke-Rockafellar directional derivative.

Definition 2.1

(Rockafellar [25])

The lsc function \(f:X\rightarrow\mathbb{R}\cup\{+\infty\}\) is directionally Lipschitzian at x̄ if for some \(y\in X\),

$$ \mathop{\limsup_{x' \stackrel{f}{\rightarrow} \bar{x}}}_{t\downarrow 0}\sup_{y'\rightarrow y} \frac{f(x'+ty')-f(x')}{t} < \infty. $$

The function \(f:X\rightarrow\mathbb{R}\cup\{+\infty\}\) is said to be radially nonconstant (rnc) if \(\forall x,y\in X\), \(\exists z\in (x,y)\), with \(f(z) \neq f(x)\), i.e., one cannot find any line segment on which f is constant.

Definition 2.2

(Avriel et al. [26])

The lsc function \(f:X\rightarrow\mathbb{R} \cup\{+\infty\}\) is said to be a quasiconvex function, if for any \(x,y \in X\), one has

$$ f(z)\leq\max\bigl\{ f(x),f(y)\bigr\} , \quad\forall x,y \in X, z\in[x,y], $$

where \([x,y]=\{x+t(y-x) : t\in(0,1)\}\).

Definition 2.3

(Clarke [27])

The lsc function \(f:X\rightarrow\mathbb{R} \cup\{+\infty\}\) is said to be a convex function at \(\bar{x}\in X\), if, for all \(x\in X\),

$$ f(x)\geq f(\bar{x})+ \langle\xi, x-\bar{x} \rangle,\quad \forall\xi\in \partial_{c} f(\bar{x}). $$

Definition 2.4

(Aussel [28])

The lsc function \(f:X\rightarrow\mathbb{R} \cup\{+\infty\}\) is said to be pseudoconvex function at \(\bar{x}\in X\), if, for all \(x\in X\),

$$\begin{aligned}& \langle\xi, x-\bar{x} \rangle\geq0, \quad\mbox{for some } \xi\in \partial_{c} f(\bar{x}) \quad\Rightarrow\quad f(x) \geq f(\bar{x}), \\& f(x)< f(\bar{x})\quad\Rightarrow\quad \langle\xi, x-\bar{x} \rangle < 0, \quad \forall\xi\in\partial_{c} f(\bar{x}). \end{aligned}$$

Theorem 2.5

(Aussel [28])

Let \(f : X\rightarrow\mathbb{R} \cup\{+\infty\}\) be lsc, quasiconvex and rnc on a convex open set \(U \subset X\). Moreover, assume that f is finite at \(\bar{x}\in U\) and \(f^{\uparrow}(\bar{x}; 0) > -\infty\). Then, for each \(x\in U\),

$$ f(x) \leq f(\bar{x}) \quad\Rightarrow\quad\forall\xi\in\partial_{c} f(\bar{x}) : \langle\xi, x-\bar{x} \rangle\leq0. $$

Given a feasible vector z̄ for the MPEC, we define the following index sets:

$$\begin{aligned}& I_{g}:=I_{g}(\bar{z}):=\bigl\{ i=1,2,\ldots,k : g_{i}(\bar{z})=0 \bigr\} , \\& \alpha:= \alpha(\bar{z})=\bigl\{ i=1,2,\ldots,l : G_{i}( \bar{z})=0,H_{i}(\bar {z})>0 \bigr\} , \\& \beta:= \beta(\bar{z})=\bigl\{ i=1,2,\ldots,l : G_{i}( \bar{z})=0,H_{i}(\bar {z})=0 \bigr\} , \\& \gamma:= \gamma(\bar{z})=\bigl\{ i=1,2,\ldots,l :G_{i}(\bar{z})>0, H_{i}(\bar{z})=0 \bigr\} . \end{aligned}$$

The set β is known as a degenerate set. If β is empty, the vector z̄ is said to satisfy the strict complementarity condition. Movahedian and Nobakhtian [8] introduced a nonsmooth type of M-stationary via the Michel-Penot subdifferential for finite-dimensional spaces. Further, Movahedian and Nobakhtian [9] extend the M-stationary notion to nonsmooth MPEC in terms of the Clarke-Rockafellar subdifferential in Banach spaces. The following definition of the M-stationary point for the nonsmooth MPEC is taken from Definition 3.1 in Movahedian and Nobakhtian [9].

Definition 2.6

A feasible point z̄ of MPEC is called the Mordukhovich stationary point if there exists \(\lambda= (\lambda^{g},\lambda^{h}, \lambda^{G}, \lambda^{H})\in\mathbb{R}^{k+p+2l}\), such that the following conditions hold:

$$\begin{aligned}& 0\in\partial_{c} f(\bar{z})+\sum_{i\in I_{g}} \lambda_{i}^{g} \partial_{c} g_{i} ( \bar{z})+\sum_{i=1}^{p} \lambda_{i}^{h} \partial_{c} h_{i}( \bar{z})- \sum_{i=1}^{l}\bigl[ \lambda_{i}^{G} \partial_{c} G_{i}( \bar{z}) + \lambda _{i}^{H} \partial_{c} H_{i}(\bar{z})\bigr], \\& \lambda_{I_{g}}^{g} \geq0, \qquad \lambda_{\gamma}^{G}=0, \qquad\lambda _{\alpha}^{H}=0, \quad \mbox{either } \lambda_{i}^{G}>0, \lambda _{i}^{H}>0 \mbox{ or } \lambda_{i}^{G} \lambda_{i}^{H}=0, \forall i \in\beta. \end{aligned}$$

The following definition of the no nonzero abnormal multiplier constraint qualification for MPEC is taken from Definition 3.3 in Movahedian and Nobakhtian [9].

Definition 2.7

Let z̄ be a feasible point of MPEC. We say that the No Nonzero Abnormal Multiplier Constraint Qualification (NNAMCQ) is satisfied at z̄ if there is no nonzero vector \(\lambda=(\lambda ^{g}, \lambda^{h}, \lambda^{G}, \lambda^{H}) \in\mathbb{R}^{k+p+2l}\), such that

$$\begin{aligned}& 0\in\sum_{i\in I_{g}} \lambda_{i}^{g} \partial_{c} g_{i} (\bar{z})+\sum _{i=1}^{p} \lambda_{i}^{h} \partial_{c} h_{i}(\bar{z})- \sum _{i=1}^{l}\bigl[\lambda_{i}^{G} \partial_{c} G_{i}(\bar{z}) + \lambda_{i}^{H} \partial_{c} H_{i}(\bar{z})\bigr], \\& \lambda_{I_{g}}^{g} \geq0,\qquad \lambda_{\gamma}^{G}=0, \qquad \lambda _{\alpha}^{H}=0, \quad \mbox{either } \lambda_{i}^{G}>0, \lambda _{i}^{H}>0 \mbox{ or } \lambda_{i}^{G} \lambda_{i}^{H}=0, \forall i \in\beta. \end{aligned}$$

Definition 2.8

(Mordukhovich [29])

A Banach space X is Asplund, or it has the Asplund property, if every convex continuous function \(\phi: U \rightarrow\mathbb{R}\) defined on an open convex subset U of X is a Frechet differential on a dense subset of U.

In the following theorem, Movahedian and Nobakhtian [9] proved a necessary optimality condition for Lipschitz MPEC on Asplund spaces.

Theorem 2.9

Let z̄ be a local optimal point for the MPEC where X is an Asplund space and all of the functions are locally Lipschitz around z̄. Then z̄ is an M-stationary point provided that the NNAMCQ holds at z̄.

Now, divide the index sets as follows. Let

$$\begin{aligned}& J^{+}:=\bigl\{ i: \lambda_{i}^{h} > 0\bigr\} , \qquad J^{-}:=\bigl\{ i:\lambda _{i}^{h} < 0\bigr\} , \\& \beta^{+} := \bigl\{ i\in\beta: \lambda_{i}^{G} > 0,\lambda_{i}^{H} > 0 \bigr\} , \\& \beta^{+}_{G}:=\bigl\{ i \in\beta: \lambda_{i}^{G}=0, \lambda_{i}^{H} > 0 \bigr\} ,\qquad \beta^{-}_{G}:= \bigl\{ i \in\beta: \lambda_{i}^{G}=0,\lambda _{i}^{H} < 0 \bigr\} , \\& \beta^{+}_{H}:=\bigl\{ i \in\beta: \lambda_{i}^{H}=0, \lambda_{i}^{G} > 0 \bigr\} , \qquad\beta^{-}_{H}:= \bigl\{ i \in\beta: \lambda_{i}^{H}=0,\lambda _{i}^{G} < 0 \bigr\} , \\& \alpha^{+}:= \bigl\{ i\in\alpha: \lambda_{i}^{G} > 0\bigr\} ,\qquad \alpha ^{-}:= \bigl\{ i\in\alpha: \lambda_{i}^{G} < 0\bigr\} , \\& \gamma^{+}:=\bigl\{ i\in\gamma:\lambda_{i}^{H} >0\bigr\} ,\qquad \gamma^{-}:=\bigl\{ i \in\gamma: \lambda_{i}^{H} < 0\bigr\} . \end{aligned}$$

3 Duality

In this section, we formulate and study a Wolfe-type dual problem for the MPEC under the convexity assumption. A Mond-Weir-type dual problem is also formulated and studied for the MPEC under convexity and generalized convexity assumptions. The Wolfe-type dual problem is formulated as follows:

$$\operatorname{WDMPEC} (\bar{z}) \max_{u,\lambda} f(u)+ \sum _{i\in I_{g}}\lambda_{i}^{g} g_{i}(u)+ \sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(u)- \sum_{i=1}^{l}\bigl[ \lambda_{i}^{G}G_{i}(u) + \lambda_{i}^{H} H_{i}(u)\bigr] $$

subject to

$$\begin{aligned} &0\in\partial_{c} f(u)+\sum _{i\in I_{g}} \lambda_{i}^{g} \partial_{c} g_{i} (u)+\sum_{i=1}^{p} \lambda_{i}^{h} \partial_{c} h_{i}(u)- \sum_{i=1}^{l}\bigl[ \lambda_{i}^{G} \partial_{c} G_{i}(u) + \lambda_{i}^{H} \partial _{c} H_{i}(u)\bigr], \\ &\lambda_{I_{g}}^{g} \geq0, \qquad\lambda_{\gamma}^{G}=0, \qquad \lambda _{\alpha}^{H}=0, \quad\mbox{either } \lambda_{i}^{G}>0, \lambda _{i}^{H}>0 \mbox{ or } \lambda_{i}^{G} \lambda_{i}^{H}=0, \forall i \in\beta, \end{aligned}$$
(3.1)

where \(\lambda=(\lambda^{g},\lambda^{h},\lambda^{G},\lambda^{H})\in \mathbb{R}^{k+p+2l}\).

Theorem 3.1

(Weak duality)

Let z̄ be feasible for MPEC where X is a Banach space, \((u,\lambda)\) feasible for \(\operatorname{WDMPEC} (\bar{z})\), and index sets \(I_{g}\), α, β, γ defined accordingly. Suppose that f, \(g_{i}\) (\(i\in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(G_{i}\) (\(i \in \alpha^{-}\cup\beta^{-}_{H}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta ^{-}_{G}\)) are convex at u and radially nonconstant. Also, assume that \(-h_{i} \) (\(i \in J^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta ^{+}_{H}\cup\beta^{+}\)), \(-H_{i}\) (\(i \in\gamma^{+}\cup\beta ^{+}_{G}\cup\beta^{+}\)) are directionally Lipschitzian, convex at u, and radially nonconstant. If \(\alpha^{-}\cup\gamma^{-} \cup\beta _{G}^{-}\cup\beta_{H}^{-} = \phi\), then, for any z feasible for the MPEC, we have

$$f(z)\geq f(u)+ \sum_{i\in I_{g}}\lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(u)- \sum _{i=1}^{l}\bigl[\lambda_{i}^{G}G_{i}(u) + \lambda _{i}^{H} H_{i}(u)\bigr]. $$

Proof

Let z be any feasible point for MPEC. Then we have

$$g_{i}(z) \leq0,\quad \forall i \in I_{g} \mbox{ and } h_{i}(z)=0, i=1,2,\ldots,p. $$

Since f is convex at u,

$$ f(z)-f(u)\geq \langle\xi, z-u \rangle, \quad\forall \xi\in \partial_{c} f(u). $$
(3.2)

Similarly, we have

$$\begin{aligned}& g_{i}(z)-g_{i}(u) \geq \bigl\langle \xi_{i}^{g}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{g}\in \partial_{c} g_{i}(u), \forall i \in I_{g}, \end{aligned}$$
(3.3)
$$\begin{aligned}& h_{i}(z)-h_{i}(u)\geq \bigl\langle \xi_{i}^{h}, z-u \bigr\rangle , \quad \forall \xi_{i}^{h} \in \partial_{c} h_{i}(u), \forall i \in J^{+}, \end{aligned}$$
(3.4)
$$\begin{aligned}& -h_{i}(z)+h_{i}(u)\geq- \bigl\langle \xi_{i}^{h}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{h}\in \partial_{c} h_{i}(u), \forall i \in J^{-}, \end{aligned}$$
(3.5)
$$\begin{aligned}& -G_{i}(z)+G_{i}(u)\geq- \bigl\langle \xi_{i}^{G}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{G}\in \partial_{c} G_{i}(u), \forall i\in\alpha^{+} \cup \beta^{+}_{H}\cup\beta^{+}, \end{aligned}$$
(3.6)
$$\begin{aligned}& -H_{i}(z)+H_{i}(u)\geq- \bigl\langle \xi_{i}^{H}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{H}\in \partial_{c} H_{i}(u), \forall i \in\gamma^{+} \cup \beta^{+}_{G}\cup\beta^{+}. \end{aligned}$$
(3.7)

If \(\alpha^{-} \cup\gamma^{-}\cup\beta_{G}^{-} \cup\beta_{H}^{-}= \phi\), multiplying (3.3)-(3.7) by \(\lambda_{i}^{g}\geq 0 \) (\(i\in I_{g}\)), \(\lambda_{i}^{h}>0\) (\(i \in J^{+}\)), \(-\lambda_{i}^{h}>0\) (\(i \in J^{-}\)), \(\lambda_{i}^{G}>0\) (\(i \in\alpha^{+} \cup\beta_{H}^{+} \cup\beta^{+}\)), \(\lambda_{i}^{H}>0\) (\(i \in\gamma^{+} \cup\beta _{G}^{+} \cup\beta^{+}\)), respectively, and adding (3.2)-(3.7), we get

$$\begin{aligned} &f(z)-f(u)+\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(z)-\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(z)-\sum _{i=1}^{p} \lambda_{i}^{h} h_{i}(u)-\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(z) \\ &\qquad{}+\sum_{i=1}^{l} \lambda_{i}^{G}G_{i}(u)-\sum _{i=1}^{l}\lambda_{i}^{H} H_{i}(z)+\sum_{i=1}^{l} \lambda_{i}^{H} H_{i}(u) \\ &\quad \geq \Biggl\langle \xi+ \sum_{i\in I_{g}} \lambda_{i}^{g} \xi_{i}^{g}+\sum _{i=1}^{p} \lambda_{i}^{h} \xi_{i}^{h}-\sum_{i=1}^{l} \bigl[\lambda_{i}^{G}\xi _{i}^{G}+ \lambda_{i}^{H} \xi_{i}^{H}\bigr],z-u \Biggr\rangle . \end{aligned}$$

From (3.1), there exist \(\bar{\xi}\in \partial_{c} f(u)\), \(\bar{\xi }_{i}^{g}\in\partial_{c} g_{i}(u)\), \(\bar{\xi}_{i}^{h}\in\partial_{c} h_{i}(u)\), \(\bar{\xi }_{i}^{G}\in\partial_{c} G_{i}(u)\), and \(\bar{\xi}_{i}^{H}\in\partial_{c} H_{i}(u)\), such that

$$\bar{\xi} + \sum_{i\in I_{g}} \lambda_{i}^{g} \bar{\xi}_{i}^{g}+\sum_{i=1}^{p} \lambda_{i}^{h} \bar{\xi}_{i}^{h}-\sum _{i=1}^{l}\bigl[\lambda _{i}^{G} \bar{\xi}_{i}^{G}+\lambda_{i}^{H} \bar{ \xi}_{i}^{H}\bigr]=0. $$

So,

$$\begin{aligned} &f(z)-f(u)+\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(z)-\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(z)-\sum _{i=1}^{p} \lambda_{i}^{h} h_{i}(u) \\ &\quad{}-\sum_{i=1}^{l} \lambda_{i}^{G}G_{i}(z)+\sum _{i=1}^{l}\lambda _{i}^{G}G_{i}(u)- \sum_{i=1}^{l}\lambda_{i}^{H} H_{i}(z)+\sum_{i=1}^{l} \lambda_{i}^{H} H_{i}(u) \geq0. \end{aligned}$$

Now, using the feasibility of z for MPEC, that is, \(g_{i}(z)\leq0\), \(h_{i}(z)=0\), \(G_{i}(z)\geq0\), \(H_{i}(z) \geq0\), we get

$$f(z)-f(u)-\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)-\sum_{i=1}^{p} \lambda _{i}^{h} h_{i}(u)+\sum _{i=1}^{l}\lambda_{i}^{G}G_{i}(u) + \sum_{i=1}^{l} \lambda_{i}^{H} H_{i}(u)\geq0. $$

Hence,

$$f(z)\geq f(u)+ \sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(u)- \sum _{i=1}^{l}\Biggl[\lambda_{i}^{G}G_{i}(u) + \sum_{i=1}^{l} \lambda_{i}^{H} H_{i}(u)\Biggr]. $$

This completes the proof. □

The following corollary is a direct consequence of Theorem 3.1.

Corollary 3.2

Let z̄ be feasible for MPEC where all constraint functions \(g_{i}\), \(h_{i}\), \(G_{i}\), \(H_{i}\) are affine and index sets \(I_{g}\), α, β, γ defined accordingly. Then, for any z feasible for the MPEC and \((u,\lambda)\) feasible for \(\operatorname{WDMPEC} (\bar{z})\), we have

$$f(z)\geq f(u)+ \sum_{i\in I_{g}}\lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(u)- \sum _{i=1}^{l}\bigl[\lambda_{i}^{G}G_{i}(u) + \lambda _{i}^{H} H_{i}(u)\bigr]. $$

Analogously, we have the following result for Asplund spaces.

Theorem 3.3

(Weak duality)

Let z̄ be feasible for MPEC where X is an Asplund space, \((u,\lambda)\) feasible for \(\operatorname{WDMPEC} (\bar{z})\) and index sets \(I_{g}\), α, β, γ defined accordingly. Suppose that f, \(g_{i} \) (\(i\in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(G_{i}\) (\(i \in \alpha^{-}\cup\beta^{-}_{H}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta ^{-}_{G}\)) are convex at u and radially nonconstant. Also, assume that \(-h_{i} \) (\(i \in J^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta ^{+}_{H}\cup\beta^{+}\)), \(-H_{i}\) (\(i \in\gamma^{+}\cup\beta ^{+}_{G}\cup\beta^{+}\)) are directionally Lipschitzian, convex at u, and radially nonconstant. If \(\alpha^{-}\cup\gamma^{-} \cup\beta _{G}^{-}\cup\beta_{H}^{-} = \phi\), then, for any z feasible for the MPEC, we have

$$f(z)\geq f(u)+ \sum_{i\in I_{g}}\lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(u)- \sum _{i=1}^{l}\bigl[\lambda_{i}^{G}G_{i}(u) + \lambda _{i}^{H} H_{i}(u)\bigr]. $$

Proof

The proof follows the lines of the proof of Theorem 3.1. □

Theorem 3.4

(Strong duality)

Assume z̄ is a locally optimal solution of MPEC where X is an Asplund space, such that NNAMCQ is satisfied at z̄ and the index sets \(I_{g}\), α, β, γ are defined accordingly. Let f, \(g_{i} \) (\(i \in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(-h_{i}\) (\(i \in J^{-}\)), \(G_{i}\) (\(i \in\alpha^{-} \cup\beta_{H}^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup \beta_{H}^{+}\cup\beta^{+}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta _{G}^{-}\)), \(-H_{i}\) (\(i \in\gamma^{+} \cup\beta_{G}^{+} \cup\beta^{+}\)) satisfy the assumption of the Theorem  3.3. Then there exists λ̄, such that \((\bar{z},\bar{\lambda})\) is an optimal solution of \(\operatorname{WDMPEC} (\bar{z})\) and the respective objective values are equal.

Proof

Since z̄ is a locally optimal solution of MPEC and the NNAMCQ is satisfied at z̄, hence, by Theorem 2.9, \(\exists \bar {\lambda}=(\bar{\lambda}^{g},\bar{\lambda}^{h},\bar{\lambda}^{G},\bar {\lambda}^{H}) \in\mathbb{R}^{k+p+2l}\), such that the nonsmooth M-stationarity conditions for MPEC are satisfied, that is, there exist \(\bar{\xi}\in \partial_{c} f(\bar{z})\), \(\bar{\xi}_{i}^{g}\in\partial_{c} g_{i}(\bar{z})\), \(\bar{\xi}_{i}^{h}\in\partial_{c} h_{i}(\bar{z})\), \(\bar{\xi }_{i}^{G}\in\partial_{c} G_{i}(\bar{z})\), and \(\bar{\xi}_{i}^{H}\in\partial_{c} H_{i}(\bar{z})\), such that

$$\begin{aligned}& 0=\bar{\xi} + \sum_{i\in I_{g}} \bar{\lambda}_{i}^{g} \bar{\xi}_{i}^{g}+\sum_{i=1}^{p} \bar{\lambda}_{i}^{h} \bar{\xi}_{i}^{h}- \sum_{i=1}^{l}\bigl[\bar{\lambda }_{i}^{G}\bar{\xi}_{i}^{G}+\bar{ \lambda}_{i}^{H} \bar{\xi}_{i}^{H} \bigr], \\& \bar{\lambda} _{I_{g}}^{g} \geq0, \qquad \bar{ \lambda}_{\gamma}^{G}=0, \bar{\lambda}_{\alpha}^{H}=0, \quad \mbox{either } \bar{\lambda }_{i}^{G}>0, \bar{ \lambda}_{i}^{H}>0 \mbox{ or } \bar{\lambda }_{i}^{G} \bar{\lambda}_{i}^{H}=0, \forall i \in\beta. \end{aligned}$$

Therefore, \((\bar{z},\bar{\lambda})\) is feasible for \(\operatorname{WDMPEC} (\bar{z})\). By Theorem 3.3, we have

$$ f(\bar{z})\geq f(u)+ \sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum _{i=1}^{p} \lambda_{i}^{h} h_{i}(u)- \sum_{i=1}^{l}\bigl[ \lambda _{i}^{G}G_{i}(u) + \lambda_{i}^{H} H_{i}(u)\bigr], $$
(3.8)

for any feasible solution \((u,\lambda)\) for \(\operatorname{WDMPEC} (\bar{z})\). Also, from the feasibility condition of MPEC and \(\operatorname{WDMPEC} (\bar{z})\), that is, for \(i \in I_{g}(\bar{z})\), \(g_{i}(\bar{z})=0\), and \(h_{i}(\bar{z})=0\), \(G_{i}(\bar{z})=0\), \(\forall i\in\alpha\cup\beta\), \(H_{i}(\bar{z})=0\), \(\forall i \in\beta\cup\gamma\), we have

$$ f(\bar{z})=f(\bar{z})+\sum_{i\in I_{g}} \bar{\lambda}_{i}^{g} g_{i}(\bar {z})+\sum _{i=1}^{p} \bar{\lambda}_{i}^{h} h_{i}(\bar{z})- \sum_{i=1}^{l} \bigl[\bar{\lambda}_{i}^{G} G_{i}(\bar{z}) + \bar{\lambda}_{i}^{H} H_{i}(\bar{z})\bigr]. $$
(3.9)

Using (3.8) and (3.9), we have

$$\begin{aligned} &f(\bar{z})+\sum_{i\in I_{g}} \bar{\lambda}_{i}^{g} g_{i}(\bar{z})+\sum_{i=1}^{p} \bar{\lambda}_{i}^{h} h_{i}(\bar{z})- \sum _{i=1}^{l}\bigl[\bar {\lambda}_{i}^{G} G_{i}(\bar{z}) + \bar{\lambda}_{i}^{H} H_{i}(\bar{z})\bigr] \\ &\quad\geq f(u)+ \sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda _{i}^{h} h_{i}(u)- \sum _{i=1}^{l}\bigl[\lambda_{i}^{G}G_{i}(u) + \lambda _{i}^{H} H_{i}(u)\bigr]. \end{aligned}$$

Hence, \((\bar{z},\bar{\lambda})\) is an optimal solution for \(\operatorname{WDMPEC} (\bar {z})\) and the respective objective values are equal. □

Example 3.1

Consider the following MPEC in \(\mathbb{R}^{2}\):

$$\begin{aligned} \operatorname{MPEC}(1) \quad \min & |z_{1}|+z_{2}^{2} \\ \mbox{subject to}: &|z_{1}|+z_{2}\geq0, \\ &-z_{2}\geq0, \\ &z_{2}\bigl(|z_{1}|+z_{2}\bigr)=0. \end{aligned}$$

Now, we formulate Wolfe-type dual problem \(\operatorname{WDMPEC} (\bar{z})\) for MPEC(1):

$$\max_{u,\lambda} |u_{1}|+u_{2}^{2}- \bigl[\lambda^{G}\bigl(|u_{1}|+u_{2}\bigr) + \lambda ^{H}(-u_{2})\bigr] $$

subject to

$$\begin{pmatrix} 0\\ 0 \end{pmatrix}= \begin{pmatrix} \xi\\ 2u_{2} \end{pmatrix} -\lambda^{G} \begin{pmatrix} \eta\\ 1 \end{pmatrix} -\lambda^{H} \begin{pmatrix} 0\\ -1 \end{pmatrix}, $$

where \(\xi,\eta\in[-1,1]\).

If β is non-empty, then either

$$\lambda^{G}>0, \qquad \lambda^{H}>0, \quad \mbox{or} \quad \lambda^{G} \lambda^{H}=0. $$

If we take the point \(\bar{z}= (0,0 )\) from the feasible region, then the index sets \(\alpha (0,0 )\) and \(\gamma (0,0 )\) are empty sets, but \(\beta:=\beta (0,0 )\) is non-empty. Also, from solving a constraint equation in the feasible region of \(\operatorname{WDMPEC} (0,0 )\), we get \(\lambda^{G}=\frac{\xi}{\eta }\) and \(\lambda^{H}=\frac{\xi}{\eta}-2u_{2}\), where \(\eta\neq0\). Since β is non-empty, we consider a \(\beta^{+}\), \(\beta^{+}_{G}\), \(\beta ^{+}_{H}\) to decide the feasible region of WDMPEC\((0,0 )\). It is clear that the assumptions of Theorem 3.1 are satisfied, so Theorem 3.1 holds between MPEC(1) and \(\operatorname{WDMPEC} (0,0 )\).

It is clear that \(\bar{z}= (0,0 )\) is the optimal solution of MPEC(1) and NNAMCQ is satisfied at z̄. Hence, the assumptions of the Theorem 3.4 are satisfied. Then, by Theorem 3.4, there exists λ̄ such that \((\bar{z},\bar{\lambda})\) is an optimal solution of \(\operatorname{WDMPEC} (0,0 )\) and the respective values are equal.

We now prove the duality relation between the mathematical programming problem with equilibrium constraints (MPEC) and the following Mond-Weir-type dual problem

$$\operatorname{MWDMPEC}(\bar{z}) \max_{u,\lambda} f(u) $$

subject to

$$\begin{aligned} &0\in\partial_{c} f(u)+\sum _{i\in I_{g}} \lambda_{i}^{g} \partial_{c} g_{i} (u)+\sum_{i=1}^{p} \lambda_{i}^{h} \partial_{c} h_{i}(u)- \sum_{i=1}^{l}\bigl[ \lambda_{i}^{G} \partial_{c} G_{i}(u) + \lambda_{i}^{H} \partial _{c} H_{i}(u)\bigr], \\ &g_{i}(u) \geq0\quad (i\in I_{g}),\qquad h_{i}(u)=0 \quad(i=1,\ldots,p), \\ &G_{i}(u) \leq0\quad (i\in\alpha\cup\beta),\qquad H_{i}(u) \leq0 \quad(i\in \beta\cup\gamma), \\ &\lambda_{I_{g}}^{g} \geq0,\qquad \lambda_{\gamma}^{G}=0, \qquad\lambda _{\alpha}^{H}=0, \quad\mbox{either } \lambda_{i}^{G}>0, \lambda _{i}^{H}>0 \mbox{ or } \lambda_{i}^{G} \lambda_{i}^{H}=0, \forall i \in\beta, \end{aligned}$$
(3.10)

where \(\lambda=(\lambda^{g},\lambda^{h},\lambda^{G},\lambda^{H})\in \mathbb{R}^{k+p+2l}\).

Theorem 3.5

(Weak duality)

Let z̄ be feasible for MPEC where X is a Banach space, \((u,\lambda)\) be feasible for \(\operatorname{MWDMPEC} (\bar{z})\), and the index sets \(I_{g}\), α, β, γ are defined accordingly. Suppose that f, \(g_{i} \) (\(i\in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(G_{i}\) (\(i \in\alpha^{-}\cup \beta^{-}_{H}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta^{-}_{G}\)) are convex at u and radially nonconstant. Also, assume that \(-h_{i} \) (\(i \in J^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta^{+}_{H}\cup\beta ^{+}\)), \(-H_{i}\) (\(i \in\gamma^{+}\cup\beta^{+}_{G}\cup\beta^{+}\)) are directionally Lipschitzian, convex at u, and radially nonconstant. If \(\alpha^{-}\cup\gamma^{-} \cup\beta_{G}^{-}\cup\beta_{H}^{-} = \phi \), then, for any z feasible for the MPEC, we have

$$f(z)\geq f(u). $$

Proof

Since f is convex at u,

$$ f(z)-f(u)\geq \langle\xi, z-u \rangle, \quad \forall \xi\in \partial_{c} f(u). $$
(3.11)

Similarly, we have

$$\begin{aligned}& g_{i}(z)-g_{i}(u) \geq \bigl\langle \xi_{i}^{g}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{g}\in \partial_{c} g_{i}(u), \forall i \in I_{g}, \end{aligned}$$
(3.12)
$$\begin{aligned}& h_{i}(z)-h_{i}(u)\geq \bigl\langle \xi_{i}^{h}, z-u \bigr\rangle , \quad \forall \xi_{i}^{h} \in \partial_{c} h_{i}(u), \forall i \in J^{+}, \end{aligned}$$
(3.13)
$$\begin{aligned}& -h_{i}(z)+h_{i}(u)\geq- \bigl\langle \xi_{i}^{h}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{h}\in \partial_{c} h_{i}(u), \forall i \in J^{-}, \end{aligned}$$
(3.14)
$$\begin{aligned}& -G_{i}(z)+G_{i}(u)\geq- \bigl\langle \xi_{i}^{G}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{G}\in \partial_{c} G_{i}(u), \forall i\in\alpha^{+} \cup \beta^{+}_{H}\cup\beta^{+}, \end{aligned}$$
(3.15)
$$\begin{aligned}& -H_{i}(z)+H_{i}(u)\geq- \bigl\langle \xi_{i}^{H}, z-u \bigr\rangle ,\quad \forall \xi_{i}^{H}\in \partial_{c} H_{i}(u), \forall i \in\gamma^{+} \cup \beta^{+}_{G}\cup\beta^{+}. \end{aligned}$$
(3.16)

If \(\alpha^{-} \cup\gamma^{-}\cup\beta_{G}^{-} \cup\beta_{H}^{-}= \phi\), multiplying (3.12)-(3.16) by \(\lambda_{i}^{g}\geq 0 \) (\(i\in I_{g}\)), \(\lambda_{i}^{h}>0\) (\(i \in J^{+}\)), \(-\lambda_{i}^{h}>0\) (\(i \in J^{-}\)), \(\lambda_{i}^{G}>0\) (\(i \in\alpha^{+} \cup\beta_{H}^{+} \cup\beta^{+}\)), \(\lambda_{i}^{H}>0\) (\(i \in\gamma^{+} \cup\beta _{G}^{+} \cup\beta^{+}\)), respectively, and adding (3.11)-(3.16), we get

$$\begin{aligned} &f(z)-f(u)+\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(z)-\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(z)-\sum _{i=1}^{p} \lambda_{i}^{h} h_{i}(u)-\sum_{i=1}^{l}\lambda _{i}^{G}G_{i}(z) \\ &\qquad{}+\sum_{i=1}^{l} \lambda_{i}^{G}G_{i}(u)-\sum _{i=1}^{l}\lambda_{i}^{H} H_{i}(z)+\sum_{i=1}^{l} \lambda_{i}^{H} H_{i}(u) \\ &\quad \geq \Biggl\langle \xi+ \sum_{i\in I_{g}} \lambda_{i}^{g} \xi_{i}^{g}+\sum _{i=1}^{p} \lambda_{i}^{h} \xi_{i}^{h}-\sum_{i=1}^{l} \bigl[\lambda_{i}^{G}\xi _{i}^{G}+ \lambda_{i}^{H} \xi_{i}^{H}\bigr],z-u \Biggr\rangle . \end{aligned}$$

From (3.10), there exist \(\bar{\xi}\in \partial_{c} f(u)\), \(\bar{\xi }_{i}^{g}\in\partial_{c} g_{i}(u)\), \(\bar{\xi}_{i}^{h}\in\partial_{c} h_{i}(u)\), \(\bar{\xi }_{i}^{G}\in\partial_{c} G_{i}(u)\), and \(\bar{\xi}_{i}^{H}\in\partial_{c} H_{i}(u)\), such that

$$\bar{\xi} + \sum_{i\in I_{g}} \lambda_{i}^{g} \bar{\xi}_{i}^{g}+\sum_{i=1}^{p} \lambda_{i}^{h} \bar{\xi}_{i}^{h}-\sum _{i=1}^{l}\bigl[\lambda _{i}^{G} \bar{\xi}_{i}^{G}+\lambda_{i}^{H} \bar{ \xi}_{i}^{H}\bigr]=0. $$

So,

$$\begin{aligned} &f(z)-f(u)+\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(z)-\sum_{i\in I_{g}} \lambda_{i}^{g} g_{i}(u)+\sum_{i=1}^{p} \lambda_{i}^{h} h_{i}(z)-\sum _{i=1}^{p} \lambda_{i}^{h} h_{i}(u) \\ &\quad{}-\sum_{i=1}^{l} \lambda_{i}^{G}G_{i}(z)+\sum _{i=1}^{l}\lambda _{i}^{G}G_{i}(u)- \sum_{i=1}^{l}\lambda_{i}^{H} H_{i}(z)+\sum_{i=1}^{l} \lambda_{i}^{H} H_{i}(u) \geq0. \end{aligned}$$

Now, using the feasibility of z and u for MPEC and \(\operatorname{MWDMPEC}(\bar {z})\), respectively, we get

$$f(z) \geq f(u). $$

This completes the proof. □

The following corollary is a direct consequence of Theorem 3.5.

Corollary 3.6

Let z̄ be feasible for MPEC where all constraint functions \(g_{i}\), \(h_{i}\), \(G_{i}\), \(H_{i}\) are affine and the index sets \(I_{g}\), α, β, γ defined accordingly. Then, for any z feasible for the MPEC and \((u,\lambda)\) feasible for \(\operatorname{MWDMPEC} (\bar{z})\), we have

$$f(z)\geq f(u). $$

Analogously, we have the following result for Asplund spaces.

Theorem 3.7

(Weak duality)

Let z̄ be feasible for MPEC where X is an Asplund space, \((u,\lambda)\) be feasible for \(\operatorname{MWDMPEC} (\bar{z})\) and the index sets \(I_{g}\), α, β, γ are defined accordingly. Suppose that f, \(g_{i}\) (\(i\in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(G_{i}\) (\(i \in\alpha^{-}\cup \beta^{-}_{H}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta^{-}_{G}\)) are convex at u and radially nonconstant. Also, assume that \(-h_{i}\) (\(i \in J^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta^{+}_{H}\cup\beta ^{+}\)), \(-H_{i}\) (\(i \in\gamma^{+}\cup\beta^{+}_{G}\cup\beta^{+}\)) are directionally Lipschitzian, convex at u, and radially nonconstant. If \(\alpha^{-}\cup\gamma^{-} \cup\beta_{G}^{-}\cup\beta_{H}^{-} = \phi \), then, for any z feasible for the MPEC, we have

$$f(z)\geq f(u). $$

Proof

The proof follows the lines of the proof of Theorem 3.5. □

Theorem 3.8

(Strong duality)

Assume z̄ is a locally optimal solution of MPEC where X is an Asplund space, such that NNAMCQ is satisfied at z̄ and the index sets \(I_{g}\), α, β, γ defined accordingly. Let f, \(g_{i}\) (\(i \in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(-h_{i}\) (\(i \in J^{-}\)), \(G_{i}\) (\(i \in\alpha^{-} \cup\beta_{H}^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta _{H}^{+}\cup\beta^{+}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta _{G}^{-}\)), \(-H_{i}\) (\(i \in\gamma^{+} \cup\beta_{G}^{+} \cup\beta^{+}\)) satisfy the assumption of the Theorem  3.7. Then there exists λ̄, such that \((\bar{z},\bar{\lambda})\) is an optimal solution of \(\operatorname{MWDMPEC} (\bar{z})\), and the respective objective values are equal.

Proof

z̄ is a locally optimal solution of MPEC and the NNAMCQ is satisfied at z̄, by Theorem 2.9, \(\exists \bar{\lambda }=(\bar{\lambda}^{g},\bar{\lambda}^{h},\bar{\lambda}^{G},\bar{\lambda }^{H}) \in\mathbb{R}^{k+p+2l}\), such that the nonsmooth M-stationarity conditions for MPEC are satisfied, that is, there exist \(\bar{\xi}\in \partial_{c} f(\bar{z})\), \(\bar{\xi}_{i}^{g}\in\partial_{c} g_{i}(\bar{z})\), \(\bar {\xi}_{i}^{h}\in\partial_{c} h_{i}(\bar{z})\), \(\bar{\xi}_{i}^{G}\in\partial_{c} G_{i}(\bar{z})\) and \(\bar{\xi}_{i}^{H}\in\partial_{c} H_{i}(\bar{z})\), such that

$$\begin{aligned}& 0=\bar{\xi} + \sum_{i\in I_{g}} \bar{\lambda}_{i}^{g} \bar{\xi}_{i}^{g}+\sum_{i=1}^{p} \bar{\lambda}_{i}^{h} \bar{\xi}_{i}^{h}- \sum_{i=1}^{l}\bigl[\bar{\lambda }_{i}^{G}\bar{\xi}_{i}^{G}+\bar{ \lambda}_{i}^{H} \bar{\xi}_{i}^{H} \bigr], \\& \bar{\lambda} _{I_{g}}^{g} \geq0, \qquad \bar{ \lambda}_{\gamma}^{G}=0,\qquad \bar{\lambda}_{\alpha}^{H}=0, \quad \mbox{either } \bar{\lambda }_{i}^{G}>0, \bar{ \lambda}_{i}^{H}>0 \mbox{ or } \bar{\lambda }_{i}^{G} \bar{\lambda}_{i}^{H}=0, \forall i \in\beta. \end{aligned}$$

Since z̄ is an optimal solution for MPEC, we have

$$\sum_{i\in I_{g}}\bar{ \lambda_{i}^{g}} g_{i}(\bar{z})=0, \qquad\sum_{i=1}^{p} \bar {\lambda_{i}^{h}} h_{i}(\bar{z})=0,\qquad \sum_{i=1}^{l}\bar{\lambda_{i}^{G}} G_{i}(\bar{z})=0,\qquad \sum_{i=1}^{l} \bar{\lambda_{i}^{H}} H_{i}(\bar{z})=0. $$

Therefore, \((\bar{z},\bar{\lambda})\) is feasible for \(\operatorname{MWDMPEC}(\bar {z})\). Also, by Theorem 3.7, for any feasible \((u,\lambda)\), we have

$$f(\bar{z}) \geq f(u). $$

Thus, \((\bar{z},\bar{\lambda})\) is an optimal solution for \(\operatorname{MWDMPEC} (\bar{z})\) and the respective objective values are equal. This completes the proof. □

Example 3.2

Consider the following MPEC problem in \(\mathbb{R}^{2}\):

$$\begin{aligned} \operatorname{MPEC} \quad \min &|z_{1}|+z_{2} \\ \mbox{subject to } & |z_{1}|+z_{2}\geq0, \\ &z_{2}-|z_{1}|\geq0, \\ &\bigl(|z_{1}|+z_{2}\bigr) \bigl(z_{2}-|z_{1}|\bigr)=0. \end{aligned}$$

The Mond-Weir-type dual problem \(\operatorname{MWDMPEC}(\bar{z})\) for the MPEC is

$$\max_{u,\lambda} |u_{1}|+u_{2} $$

subject to

$$ \begin{pmatrix} 0\\ 0 \end{pmatrix}= \begin{pmatrix} \xi_{1}\\ 1 \end{pmatrix} -\lambda^{G} \begin{pmatrix} \xi_{2}\\ 1 \end{pmatrix} - \lambda^{H} \begin{pmatrix} \eta\\ 1 \end{pmatrix}, $$
(3.17)

where \(\xi_{1},\xi_{2},\eta\in[-1,1]\),

$$\begin{aligned}& \lambda^{G} \bigl(|u_{1}|+u_{2} \bigr) \leq0, \end{aligned}$$
(3.18)
$$\begin{aligned}& \lambda^{H} \bigl(u_{2}-|u_{1}| \bigr) \leq0, \end{aligned}$$
(3.19)

if β is non-empty, then either

$$\lambda^{G}>0, \qquad \lambda^{H}>0, \quad\mbox{or} \quad \lambda^{G} \lambda^{H}=0. $$

From (3.17), \(\lambda^{G}\xi_{2}+\lambda^{H}\eta=\xi_{1}\), and \(\lambda^{G}+\lambda^{H}=1\), we get \(\lambda^{H}=\frac{\xi_{2}-\xi_{1}}{\xi_{2}-\eta}\) and \(\lambda^{G}=\frac{\xi _{1}-\eta}{\xi_{2}-\eta}\), where \(\xi_{2}\neq\eta\). If \(\bar{z}= (0,0 )\), then the index sets \(\alpha (0,0 )\) and \(\gamma (0,0 )\) are empty sets, but \(\beta (0,0 )\) is non-empty. It is clear that the assumptions of Corollary 3.6 are satisfied. So, Corollary 3.6 holds between MPEC and \(\operatorname{MWDMPEC} (0,0 )\).

Also, we can see that the NNAMCQ is satisfied at z̄. Then by Theorem 3.8 there exists \(\bar{\lambda}=(\bar{\lambda}^{G}, \bar{\lambda }^{H})\) such that \((\bar{z},\bar{\lambda})\) is an optimal solution of \(\operatorname{MWDMPEC} (0,0 )\) and the optimal values are equal.

Now, we establish weak and strong duality theorems for the MPEC and its Mond-Weir-type dual problem under generalized convexity assumptions.

Theorem 3.9

(Weak duality)

Let z̄ be feasible for MPEC where X is a Banach space, \((u,\lambda)\) be feasible for \(\operatorname{MWDMPEC} (\bar{z})\), and the index sets \(I_{g}\), α, β, γ are defined accordingly. Suppose that f is pseudoconvex at z̄, \(g_{i}\) (\(i\in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(G_{i}\) (\(i \in\alpha^{-}\cup\beta^{-}_{H}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta^{-}_{G}\)) are quasiconvex at u and radially nonconstant. Also, assume that \(-h_{i} \) (\(i \in J^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta^{+}_{H}\cup\beta^{+}\)), \(-H_{i}\) (\(i \in\gamma^{+}\cup\beta ^{+}_{G}\cup\beta^{+}\)) are directionally Lipschitzian, quasiconvex at u, and radially nonconstant. If \(\alpha^{-}\cup\gamma^{-} \cup\beta _{G}^{-}\cup\beta_{H}^{-} = \phi\), then, for any z feasible for the MPEC, we have

$$f(z)\geq f(u). $$

Proof

Suppose that, for some feasible point z, such that \(f(z)< f(u)\), then, by pseudoconvexity of f at u, we have

$$ \langle\xi, z-u \rangle< 0,\quad \forall \xi\in \partial_{c} f(u). $$
(3.20)

From (3.10), there exist \(\bar{\xi}_{i}^{g}\in\partial_{c} g_{i}(u)\) (\(i\in I_{g}\)), \(\bar{\xi}_{i}^{h}\in\partial_{c} h_{i}(u)\) (\(i=1,\ldots,p\)), \(\bar{\xi }_{i}^{G}\in\partial_{c} G_{i}(u)\) (\(i \in\alpha\cup\beta\)) and \(\bar{\xi }_{i}^{H}\in\partial_{c} H_{i}(u)\) (\(i\in\beta\cup\gamma\)), such that

$$ -\sum_{i\in I_{g}}\lambda_{i}^{g} \bar{\xi}_{i}^{g}-\sum_{i=1}^{p} \lambda _{i}^{h} \bar{\xi}_{i}^{h}+ \sum_{\alpha\cup\beta}\lambda_{i}^{G}\bar{ \xi }_{i}^{G}+\sum_{\beta\cup\gamma} \lambda_{i}^{H} \bar{\xi}_{i}^{H} \in \partial_{c} f(u). $$
(3.21)

By (3.20), we get

$$ \Biggl\langle \Biggl( -\sum_{i\in I_{g}} \lambda_{i}^{g} \bar{\xi}_{i}^{g}-\sum _{i=1}^{p} \lambda_{i}^{h} \bar{\xi}_{i}^{h}+\sum_{\alpha\cup\beta} \lambda _{i}^{G}\bar{\xi}_{i}^{G}+\sum _{\beta\cup\gamma} \lambda_{i}^{H} \bar{\xi }_{i}^{H} \Biggr), z-u \Biggr\rangle < 0. $$
(3.22)

For each \(i\in I_{g}\), \(g_{i}(z)\leq0 \leq g_{i}(u)\). Hence, by Theorem 4.4 in [9], we have

$$ \langle\xi, z-u \rangle\leq0, \quad\forall \xi\in \partial_{c} g_{i}(u), \forall i \in I_{g}. $$
(3.23)

Similarly, we have

$$ \langle\xi, z-u \rangle\leq0, \quad\forall \xi\in \partial_{c} h_{i}(u), \forall i \in J^{+}. $$
(3.24)

Now, for any feasible point u of \(\operatorname{MWDMPEC} (\bar{z})\), and for each \(i \in J^{-}\), \(0=-h_{i}(u)=h_{i}(z)\). On the other hand, \(-G_{i}(z)\leq -G_{i}(u)\), \(\forall i \in\alpha^{+} \cup\beta_{H}^{+}\), and \(-H_{i}(z)\leq-H_{i}(u)\), \(\forall i \in\gamma^{+} \cup\beta_{G}^{+}\). Since all of these functions are directionally Lipschitzian, by Theorem 2.5, we get

$$\begin{aligned}& \langle\xi, z-u \rangle\geq0,\quad \forall \xi\in\partial_{c} h_{i}(u), \forall i \in J^{-}, \end{aligned}$$
(3.25)
$$\begin{aligned}& \langle\xi, z-u \rangle\geq0, \quad\forall \xi\in\partial_{c} G_{i}(u), \forall i \in\alpha^{+} \cup\beta_{H}^{+} , \end{aligned}$$
(3.26)
$$\begin{aligned}& \langle\xi, z-u \rangle\geq0, \quad\forall \xi\in\partial_{c} H_{i}(u), \forall i \in\gamma^{+} \cup\beta_{G}^{+}. \end{aligned}$$
(3.27)

From equation (3.23)-(3.27), it is clear that

$$\begin{aligned}& \bigl\langle \bar{\xi}_{i}^{g}, z-u \bigr\rangle \leq0 \quad (i\in I_{g}),\qquad \bigl\langle \bar{\xi}_{i}^{h}, z-u \bigr\rangle \leq0\quad \bigl(i\in J^{+}\bigr),\qquad \bigl\langle \bar{\xi}_{i}^{h}, z-u \bigr\rangle \geq0 \quad\bigl(i\in J^{-} \bigr), \\& \bigl\langle \bar{\xi}_{i}^{G}, z-u \bigr\rangle \geq0,\quad \forall i \in \alpha^{+} \cup\beta_{H}^{+},\qquad \bigl\langle \bar{\xi}_{i}^{H}, z-u \bigr\rangle \geq0, \quad \forall i \in\gamma^{+} \cup\beta_{G}^{+}. \end{aligned}$$

Since \(\alpha^{-}\cup\gamma^{-} \cup\beta_{G}^{-}\cup\beta_{H}^{-} = \phi\), we have

$$\begin{aligned}& \biggl\langle \sum_{\alpha\cup\beta}\lambda_{i}^{G} \bar{\xi}_{i}^{G}, z-u \biggr\rangle \geq0,\qquad \biggl\langle \sum_{\beta\cup\gamma} \lambda_{i}^{H} \bar{\xi}_{i}^{H}, z-u \biggr\rangle \geq0, \\& \biggl\langle \sum_{i\in I_{g}}\lambda_{i}^{g} \bar{\xi}_{i}^{g}, z-u \biggr\rangle \geq0,\qquad \Biggl\langle \sum_{i=1}^{p} \lambda_{i}^{h} \bar{\xi}_{i}^{h}, z-u \Biggr\rangle \geq0. \end{aligned}$$

Therefore,

$$\Biggl\langle \Biggl( -\sum_{i\in I_{g}} \lambda_{i}^{g} \bar{\xi}_{i}^{g}-\sum _{i=1}^{p} \lambda_{i}^{h} \bar{\xi}_{i}^{h}+\sum_{\alpha\cup\beta} \lambda _{i}^{G}\bar{\xi}_{i}^{G}+\sum _{\beta\cup\gamma} \lambda_{i}^{H} \bar{\xi }_{i}^{H} \Biggr), z-u \Biggr\rangle \geq0, $$

which contradicts (3.22). Hence, \(f(z)\geq f(u)\). This completes the proof. □

Analogously, we have the following result for Asplund spaces.

Theorem 3.10

(Weak duality)

Let z̄ be feasible for MPEC where X is an Asplund space, \((u,\lambda)\) be feasible for \(\operatorname{MWDMPEC} (\bar{z})\), and the index sets \(I_{g}\), α, β, γ are defined accordingly. Suppose that f is pseudoconvex at z̄, \(g_{i}\) (\(i\in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(G_{i}\) (\(i \in\alpha^{-}\cup\beta^{-}_{H}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta^{-}_{G}\)) are quasiconvex at u and radially nonconstant. Also, assume that \(-h_{i} \) (\(i \in J^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup\beta^{+}_{H}\cup\beta^{+}\)), \(-H_{i}\) (\(i \in\gamma^{+}\cup\beta ^{+}_{G}\cup\beta^{+}\)) are directionally Lipschitzian, quasiconvex at u, and radially nonconstant. If \(\alpha^{-}\cup\gamma^{-} \cup\beta _{G}^{-}\cup\beta_{H}^{-} = \phi\), then, for any z feasible for the MPEC, we have

$$f(z)\geq f(u). $$

Proof

The proof follows the lines of the proof of Theorem 3.9. □

Theorem 3.11

(Strong duality)

Assume z̄ is a locally optimal solution of MPEC where X is an Asplund space, such that NNAMCQ is satisfied at z̄, and the index sets \(I_{g}\), α, β, γ are defined accordingly. Let f, \(g_{i}\) (\(i \in I_{g}\)), \(h_{i}\) (\(i \in J^{+}\)), \(-h_{i}\) (\(i \in J^{-}\)), \(G_{i}\) (\(i \in\alpha^{-} \cup\beta_{H}^{-}\)), \(-G_{i}\) (\(i \in\alpha^{+} \cup \beta_{H}^{+}\cup\beta^{+}\)), \(H_{i}\) (\(i \in\gamma^{-} \cup\beta _{G}^{-}\)), \(-H_{i}\) (\(i \in\gamma^{+} \cup\beta_{G}^{+} \cup\beta^{+}\)) satisfy the assumption of Theorem  3.10. Then there exists λ̄, such that \((\bar{z},\bar{\lambda})\) is an optimal solution of \(\operatorname{MWDMPEC} (\bar{z})\), and the respective objective values are equal.

Proof

The proof follows the lines of the proof of Theorem 3.8, invoking Theorem 3.10. □

4 Results and discussion

We have studied mathematical programs with equilibrium constraints (MPECs). The objective function and functions in the constraint part are assumed to be lower semicontinuous. We studied the Wolfe-type dual problem for the MPEC under the convexity assumption. A Mond-Weir-type dual problem was also formulated and studied for the MPEC under convexity and generalized convexity assumptions. Conditions for weak duality theorems were given to relate the MPEC and two dual programs in Banach space, respectively. Also conditions for strong duality theorems were established in an Asplund space. We also discussed the cases when all the constraint functions are affine. Two numerical examples were given to illustrate the Wolfe-type duality and the Mond-Weir-type duality with our MPECs, respectively.